Average Error: 16.8 → 3.7
Time: 52.8s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\left(-R \cdot \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) + \frac{\pi}{2} \cdot R\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
\left(-R \cdot \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) + \frac{\pi}{2} \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r751911 = phi1;
        double r751912 = sin(r751911);
        double r751913 = phi2;
        double r751914 = sin(r751913);
        double r751915 = r751912 * r751914;
        double r751916 = cos(r751911);
        double r751917 = cos(r751913);
        double r751918 = r751916 * r751917;
        double r751919 = lambda1;
        double r751920 = lambda2;
        double r751921 = r751919 - r751920;
        double r751922 = cos(r751921);
        double r751923 = r751918 * r751922;
        double r751924 = r751915 + r751923;
        double r751925 = acos(r751924);
        double r751926 = R;
        double r751927 = r751925 * r751926;
        return r751927;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r751928 = R;
        double r751929 = phi2;
        double r751930 = cos(r751929);
        double r751931 = phi1;
        double r751932 = cos(r751931);
        double r751933 = r751930 * r751932;
        double r751934 = lambda1;
        double r751935 = sin(r751934);
        double r751936 = lambda2;
        double r751937 = sin(r751936);
        double r751938 = cos(r751936);
        double r751939 = cos(r751934);
        double r751940 = r751938 * r751939;
        double r751941 = fma(r751935, r751937, r751940);
        double r751942 = sin(r751929);
        double r751943 = sin(r751931);
        double r751944 = r751942 * r751943;
        double r751945 = fma(r751933, r751941, r751944);
        double r751946 = asin(r751945);
        double r751947 = r751928 * r751946;
        double r751948 = -r751947;
        double r751949 = atan2(1.0, 0.0);
        double r751950 = 2.0;
        double r751951 = r751949 / r751950;
        double r751952 = r751951 * r751928;
        double r751953 = r751948 + r751952;
        return r751953;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 16.8

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]

  2. Simplified16.8

    \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)}\]
  3. Using strategy rm

  4. Applied cos-diff3.6

    \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2}, \sin \phi_2 \cdot \sin \phi_1\right)\right)\]
  5. Using strategy rm

  6. Applied add-log-exp3.6

    \[\leadsto R \cdot \color{blue}{\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2, \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}\]

  7. Simplified3.6

    \[\leadsto R \cdot \log \color{blue}{\left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)}\]
  8. Using strategy rm

  9. Applied acos-asin3.7

    \[\leadsto R \cdot \log \left(e^{\color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}}\right)\]

  10. Applied exp-diff3.7

    \[\leadsto R \cdot \log \color{blue}{\left(\frac{e^{\frac{\pi}{2}}}{e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}}\right)}\]

  11. Applied log-div3.7

    \[\leadsto R \cdot \color{blue}{\left(\log \left(e^{\frac{\pi}{2}}\right) - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)\right)}\]

  12. Simplified3.7

    \[\leadsto R \cdot \left(\color{blue}{\frac{\pi}{2}} - \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\right)\right)\]

  13. Simplified3.7

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)\]
  14. Using strategy rm

  15. Applied sub-neg3.7

    \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} + \left(-\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)\right)}\]

  16. Applied distribute-rgt-in3.7

    \[\leadsto \color{blue}{\frac{\pi}{2} \cdot R + \left(-\sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) \cdot R}\]

  17. Final simplification3.7

    \[\leadsto \left(-R \cdot \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_2 \cdot \sin \phi_1\right)\right)\right) + \frac{\pi}{2} \cdot R\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))